Question

What happens to the x-intercepts of the function y=x2−25
when the graph is reflected over the x-axis? Explain.(1 point)
Responses

This function does not have any x-intercepts, so no changes are necessary.
This function does not have any x -intercepts, so no changes are necessary.

The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.
The x -intercepts remain the same because the y -values get negated during a reflection over the x -axis.

The x-intercepts become negated.
The x -intercepts become negated.

The x-values and y-values switch.

Answer 1

To analyze what happens to the x-intercepts of the function \( y = x^2 - 25 \) when the graph is reflected over the x-axis, we first need to understand what x-intercepts are. X-intercepts are points where the graph intersects the x-axis, which occurs when \( y = 0 \).

1. **Finding the x-intercepts of the original function**:
- Set \( y = 0 \):
\[
0 = x^2 - 25
\]
- Solve for \( x \):
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
This means the x-intercepts are at \( (5, 0) \) and \( (-5, 0) \).

2. **Reflection over the x-axis**:
- When we reflect the graph over the x-axis, the y-values are negated. Therefore, the new function after reflection would be \( y = -(x^2 - 25) = -x^2 + 25 \).
- Now, to find any new x-intercepts, we again set \( y = 0 \):
\[
0 = -x^2 + 25
\]
- Solve for \( x \):
\[
x^2 = 25
\]
\[
x = 5 \quad \text{or} \quad x = -5
\]
The x-intercepts remain the same, at \( (5, 0) \) and \( (-5, 0) \).

Thus, when the graph of \( y = x^2 - 25 \) is reflected over the x-axis, the x-intercepts remain the same.

**Correct response**: The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.